Exponential tilt - Knowlpedia

Let $X$ be a real random variable on $(\Omega,P)$ . If $M(\theta)=\mathbb E_P e^{\theta X}<\infty$ , the exponential tilt of $P$ at $\theta$ is

P_\theta(A)=\frac{\int_A e^{\theta X}\,dP}{M(\theta)}.

The normalizing factor is strictly positive, so this defines a probability measure. Its density relative to $P$ is $e^{\theta X}/M(\theta)$ .

Parameter domain

The tilt is defined only where the normalizing integral is finite. If $X$ is bounded, every real parameter is allowed. The new measure is equivalent to $P$ , since its density is everywhere positive up to the usual almost-everywhere convention. For a periodic observable with normalized angular measure, the same operation is simply a positive reweighting of that angular average.

References

Fithian, Exponential Families (derivatives of the log-partition function).